Intramolecular bottlenecks

A Zeroth-Order Calculation of the Rate Constant for Crossing Intramolecular Bottlenecks The Rate Constant for Crossing the Separatrix... 

The existence of bottlenecks to Hamiltonian transport suggests that intramolecular energy flow can be highly nonergodic. Thus, accounting for the bottlenecks should greatly improve chemical reaction rate theories. For example, for the 4 1 resonance shown in Figs. 2 and 3, the intramolecular bottleneck should be located at... 

For the case in Fig. 4 the intramolecular bottleneck is expected to be associated with... 

Figure 14. An Hel2 surface of section for an unstable trajectory which forms a collision complex. The total energy is —2661.6 cm . Also shown are the reaction separatrix and the intramolecular bottleneck, (a) Graph showing the full dynamics of the trajectory. (b)-(f) Graphs illustrating the trajectory over five consecutive time ranges. These graphs are arranged to demonstrate the manner in which the trajectory moves with respect to the bottleneck and the separatrix. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

Davis and Gray also demonstrated the existence of a series of intramolecular energy transfer bottlenecks, each corresponding to the breakup of a KAM torus. For example, for I2 in the vibrational state v = 20 they found intramolecular bottlenecks associated with frequency ratios equal to (3 + g) and up. However, Davis and Gray found that the last golden mean torus to be broken up is the most effective bottleneck to intramolecular energy transfer and is therefore... 

It is worth mentioning that Davis and Gray also found that at low energy, for example, when I2 is initially in a vibrational state with v < 5, no classical dissociation occurs. Furthermore, if I2 is initially in a vibrational state with 20 > V > 5, the dynamics appears to be so complicated that including only one intramolecular bottleneck does not suffice. Indeed, in the case of v = 10 Davis and Gray used two intramolecular bottlenecks to model the Hel2 fragmentation reaction. The two bottlenecks on a PSS are illustrated in Fig. 17. It is seen that... 

The Davis-Gray theory teaches us that by retaining the most important elements of the nonhnear reaction dynamics it is possible to accurately locate the intramolecular bottlenecks and to have an exact phase space separatrix as the transition state. Unfortunately, even for systems with only two DOFs, there may be considerable technical difficulties associated with locating the exact bottlenecks and the separatrix. Exact calculations of the fluxes across these phase space structures present more problems. For these reasons, further development of unimolecular reaction rate theory requires useful approximations. 

Zhao and Rice then developed an approximation to locate the intramolecular bottlenecks and calculate the associated flux. There are two principal motivations for the development of such an approximation. These are, first, the need to simplify the very complicated mapping-based calculation of the flux crossing a cantorus so as to make the calculation practical in systems with many DOFs and, second, the desirability of having a simple representation of the intramolecular energy transfer barrier in terms of molecular properties. 

To locate the intramolecular bottleneck, it is assumed that there is no energy transfer to the van der Waals stretching motion or to rotational motion, so the energy in all other DOFs is conserved. This energy is negative, corresponding to bounded motion, and is given by... 

To calculate the normalization constant Na in feintra Zhao and Rice proceeded as follows. Assuming that the system is prepared in a state with all the phase-space points inside the intramolecular bottleneck dividing surface, then the density of these phase-space points can be written as... 

Figure 19. A schematic plot of the ideal bottlenecks on the Poincare surface of section for van der Waals molecule predissociation. R is the van der Waals bond length and P is the conjugate momentum. 5i is the intramolecular bottleneck dividing surface and S2 is the intermoleculear bottleneck dividing surface.

When the effect of intramolecular energy transfer is taken into account, more accurate rate constants can be obtained. We first compare the rate constants associated with the intramolecular bottleneck from the MRRKM theory with those from the Davis-Gray turnstile approach. As seen in Table III, they are in reasonable agreement. Hence, the Davis-Gray theory and the MRRKM theory predict similar overall reaction rates. This is demonstrated in Table IV. Table IV also shows that the predissociation rate constants would have been overestimated by a factor more than 100 if the RRKM theory were to be directly applied. 

Intramolecular Bottleneck Predissociation Rate Constants (in cm ) for the T-Shaped HeT Molecule"... 

Clearly, the A and B isomer states should be inside the separatrix, and the state C should be in the phase-space region outside of the separatrix but inside the energy boundary. A schematic diagram of this three-state isomerization model is presented in Fig. 20. From the results of previous analyses of predissociation we expect that within the A and B domains there are, in general, intramolecular bottlenecks to energy transfer. However, these bottlenecks are... 

MRRKM theory, from RIT, from direct trajectory simulations, and, for reference purposes, from the RRKM theory. In particular, a test of the effect of the RRKM choice of transition state on the predicted rate of isomerization is made by neglecting the contribution of intramolecular energy transfer (Model No. 1). It is seen that the RRKM choice of transition state leads to considerable error the isomerization rate constant predicted is greater than those from the MRRKM theory and RIT by as much as a factor of 4. With intramolecular bottlenecks taken into account, both RIT and the MRRKM theory agree well with trajectory calculations. 

The fact that classical unstable periodic trajectories can manifest themselves in the Wigner function implies that nonstatistical behavior in the quanmm dynamics can be intimately related to the phase-space structure of the classical molecular dynamics. Consider, for example, the bottlenecks to intramolecular energy flow. Since the intramolecular bottlenecks are caused by remnants of the most robust tori, they are presumably related to the least unstable periodic trajectories. Hence quantum scars, being most significant in the case of the least unstable periodic trajectories, are expected to be more or less connected with intramolecular bottlenecks. Indeed, this observation motivated a recent proposal [75] to semiclassically locate quantum intramolecular bottlenecks. Specifically, the most robust intramolecular bottlenecks are associated with the least unstable periodic trajectories for which Eq. (332) holds, that is,... 

This hnding concerning quantum transport in classically chaotic systems sheds new light on quantum effects in unimolecular reaction dynamics. For example, one expects that intramolecular bottlenecks associated with canton, if treated quantum mechanically, would be more effective than in a classical statistical theory even when nh is smaller than the reaction flux crossing the intramolecular dividing surface. Clearly, it would be interesting to examine realistic molecular systems in a similar fashion. 

Figure 8.15 illustrates the presence of an intramolecular bottleneck in the interaction region of phase space. The transition rate through the turnstile in this bottleneck can be calculated using concepts described in section 4.3.1. An intramolecular bottleneck, such as the one depicted in figure 8.15, is expected to give rise to intrinsic non-RRKM behavior. 

---